A333322 Decimal expansion of (3/8) * sqrt(3).

6, 4, 9, 5, 1, 9, 0, 5, 2, 8, 3, 8, 3, 2, 8, 9, 8, 5, 0, 7, 2, 7, 9, 2, 3, 7, 8, 0, 6, 4, 7, 0, 2, 1, 3, 7, 6, 0, 3, 5, 5, 1, 9, 7, 0, 1, 7, 8, 8, 9, 2, 7, 3, 5, 5, 2, 0, 9, 2, 7, 6, 1, 7, 2, 9, 4, 4, 7, 4, 8, 8, 1, 3, 4, 0, 8, 0, 0, 0, 1, 3, 9, 0, 5, 4, 2, 9, 8, 2, 0, 0, 3, 3, 9, 6, 8, 2, 1, 5, 8, 7, 8, 3, 5, 9, 8, 0, 3, 0, 3, 0, 7, 7, 7, 5, 1, 3, 6, 3, 6

Offset

0,1

Comments

This is the area of the regular hexagon of diameter 1.

From Bernard Schott, Apr 09 2022 and Oct 01 2022: (Start)

For any triangle ABC, where (A,B,C) are the angles:

sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference],

cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference],

and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]:

(ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3).

The equalities are obtained only when triangle ABC is equilateral. (End)

References

O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19.

D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111.

Links

Table of n, a(n) for n=0..119.

Scott Brown, Problem 3453, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343.

Formula

Equals A104954/2 or A104956/4.

Example

0.649519052838328985...

Mathematica

RealDigits[(3/8) * Sqrt[3], 10, 120][[1]]

Prog

(PARI) sqrt(27)/8 \\ Charles R Greathouse IV, Apr 09 2022

Crossrefs

Cf. A002194 (sqrt(3)), A104954.

Cf. A010527, A020821, A104956, A152623 (other geometric inequalities).

Sequence in context: A231535 A019931 A195414 * A348896 A153630 A113276

Adjacent sequences: A333319 A333320 A333321 * A333323 A333324 A333325

Keyword

cons,nonn

Author

Kritsada Moomuang, Mar 15 2020

Status

approved